Some extremal self-dual codes and unimodular lattices in dimension 40

Abstract

In this paper, binary extremal singly even self-dual codes of length 40 and extremal odd unimodular lattices in dimension 40 are studied. We give a classification of extremal singly even self-dual codes of length 40. We also give a classification of extremal odd unimodular lattices in dimension 40 with shadows having 80 vectors of norm 2 through their relationships with extremal doubly even self-dual codes of length 40. * This work was supported by JST PRESTO program.

A projection decoding of a binary extremal self-dual code of length 40

“…An additive code C over GF (4) of length n is an additive subgroup of GF (4) n . Since C is a vector space over GF (2), it has a GF (2)-basis consisting of k (0 ≤ k ≤ 2n) vectors whose entries are in GF (4). We call C an (n, 2 k ) code.…”

“…If C is a self-dual additive (n, 2 n ) code over GF (4), let C be the binary linear [4n, n] code obtained from C by mapping each GF (4) component to a 4-tuple in GF (2) 4 as follows : 0 → 0000, 1 → 0011, ω → 0101, and ω → 0110. Let d 4 be the [4,1,4] Construction B: Assume that n is even. Define ρ B (C) = C + (d n 4 ) 0 + e B .…”

“…Shortly after, it was shown [3] that there are exactly 16470 extremal doubly-even self-dual binary codes of length 40. Then, it has been shown [4] that there are exactly 10200655 extremal singly-even self-dual codes of length 40.…”

“…For example, Pless [21] showed how to decode the extended binary Golay [24, 12,8] code by hand. The [6,3,4] hexacode over GF (4) was used here to decode the extended Golay [24, 12,8] code. Later, Gaborit-Kim-Pless [12] suggested two handy decoding algorithms for three doubly-even self-dual [32, 16,8] codes such as the binary Reed-Muller [32, 16,6] code and two other doubly-even self-dual [32, 16,8] codes denoted by C83 (or 2g 16 ) and C84 (or 8f 4 ) in the notation of [20].…”

“…In this paper, we describe two new efficient decoding algorithms for an extremal self-dual [40, 20,8] code C DE 40,1 by hand with the help of a Hermitian self-dual [10,5,4] code E 10 over GF (4). The algorithm is called the representation decoding algorithm.…”